The given expression is:

$\frac{5 \sqrt{72}}{\sqrt{24} + \sqrt{54}}$

To simplify this expression, let's break it down step-by-step.

Step 1: Simplify the Square Roots

1. Simplify $\sqrt{72}$: $\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}$

2. Simplify $\sqrt{24}$: $\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}$

3. Simplify $\sqrt{54}$: $\sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \cdot \sqrt{6} = 3\sqrt{6}$

Step 2: Substitute Back into the Expression

Now, substitute these simplified values back into the expression:

$\frac{5 \cdot 6\sqrt{2}}{2\sqrt{6} + 3\sqrt{6}}$

This simplifies to:

$\frac{30\sqrt{2}}{(2 + 3)\sqrt{6}} = \frac{30\sqrt{2}}{5\sqrt{6}}$

Step 3: Simplify the Fraction

Now, divide both the numerator and the denominator by 5:

$\frac{30\sqrt{2}}{5\sqrt{6}} = \frac{6\sqrt{2}}{\sqrt{6}}$

Step 4: Rationalize the Denominator

To remove the square root in the denominator, multiply the numerator and the denominator by $\sqrt{6}$:

$\frac{6\sqrt{2} \cdot \sqrt{6}}{\sqrt{6} \cdot \sqrt{6}} = \frac{6\sqrt{12}}{6} = \sqrt{12}$

Step 5: Simplify $\sqrt{12}$

Finally,

$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$

Final Answer

$\frac{5 \sqrt{72}}{\sqrt{24} + \sqrt{54}} = 2\sqrt{3}$

Would you like further clarification on any of these steps?

Follow-up Questions

1. How would you simplify similar expressions involving square roots?
2. What is the process of rationalizing the denominator, and why is it used?
3. Can you apply these steps to simplify $\frac{\sqrt{50}}{\sqrt{2} + \sqrt{8}}$?
4. What other methods exist for simplifying expressions with square roots in the denominator?
5. How does simplifying radicals differ when working with cube roots instead of square roots?

Tip

When simplifying square roots, look for perfect square factors (like 4, 9, 16, etc.) to make the simplification easier.